Extended Reeb graphs for surface understanding and description

The aim of this paper is to describe a conceptual model for surface representation based on topological coding, which defines a sketch of a surface usable for classification or compression purposes. Theoretical approaches based on differential topology and geometry have been used for surface coding, for example Morse theory and Reeb graphs. To use these approaches in discrete geometry, it is necessary to adapt concepts developed for smooth manifolds to discrete surface models, as for example piecewise linear approximations. A typical problem is represented by degenerate critical points, that is non-isolated critical points such as plateaux and flat areas of the surface. Methods proposed in literature either do not consider the problem or propose local adjustments of the surface, which solve the theoretical problem but may lead to a wrong interpretation of the shape by introducing artefacts, which do not correspond to any shape feature. In this paper, an Extended Reeb Graph representation (ERG) is proposed, which can handle degenerate critical points, and an algorithm is presented for its construction.